r-COSTAR MODULES - ieja.net

International Electronic Journal of Algebra
Volume 8 (2010) 167-176

r-COSTAR MODULES

Hongxing Liu and Shunhua Zhang

Received: 31 March 2010; Revised: 13 May 2010
Communicated by Abdullah Harmancı

Abstract. We give the definition of r-costar modules as a generalization of
costar modules under artin algebra situation. We also study characterizations
of r-costar modules and give characterizations of r-costar modules with some
special properties.
Mathematics Subject Classification (2000): 16D90, 16E30
Keywords: r-costar module, resolving subcategory, reflexive module

1. Introduction

The theory of equivalences and dualities between module subcategories, origi-
nating in the well-known theory of Morita theorems, had been studied extensively.
Both quasi-progenerators and tilting modules over arbitrary rings induce equiva-
lences between certain categories of modules. Menini and Orsatti introduced a gen-
eralization of these modules that have come to be called ∗-modules, cf. [5]. Namely,
let CA be a subcategory of Mod-A closed under submodules and containing AA, and
let YR be a subcategory of Mod-R closed under direct sums and epimorphic images,
then any equivalence between CA and YR is represented by a bimodule APR, via the
adjoint pair (TP , HP ). P is called ∗-module if we put A = End(PR), TP = − ⊗A P ,
HP = HomR(P, −), KA = HomR(P, Q), where Q is a cogenerator of Mod-A, then
the pair of the functors (TP , HP ) defines an equivalence between Gen(PR) and
Cogen(KA). In [3], Colpi gave that P is a ∗-module if and only if P is selfsmall
and, for any exact sequence 0 → L → M → N → 0 with M, N ∈ Gen(P ), the
induced sequence 0 → HP (L) → HP (M ) → HP (N ) → 0 is exact if and only if
L ∈ Gen(P ).

Wei introduced the notion of ∗s-modules, where s denotes static, by replacing
the subcategory Gen(P ) in the theory of ∗-module with the subcategory Stat(P )
and some results on ∗-module are successively extended to ∗s-modules, cf. [7].

Supported by NSF of China (Grant No. 10771112) NSF of Shandong Province (Grant No.
Y2008A05) and Young Scientist Reward Funding from Shandong Province(No.2008BS01016).

168 HONGXING LIU AND SHUNHUA ZHANG

On the other side, a dual notion of quasi-progenerators called quasi-duality mod-
ules and cotilting modules dual to tilting modules have been a central topic of re-
cent investigation in module theory. Colby and Fuller generalized these modules to
costar modules which may be viewed as dual to ∗-modules in a sense, cf. [1].

Inspired by [7], we shall investigate r-costar modules by replacing the subcate-
gory cogen(P) in the theory of costar modules with the subcategory Ref(PΛ) under
artin algebra situation. Some similar results are obtained.

2. Preliminaries

Let R be an associative ring with nonzero identity, Mod-R (R-Mod) denotes the
category of all right (left) R-modules, while mod-R and R-mod denote their sub-
categories of finitely generated modules. We shall let projR be the full subcategory
of all projective modules in mod-R (or R-mod). Given a module PR (or RP ), we
denote

KerExtRi≥e(−, P ) = {X ∈ mod-R(or R-mod) | ExtiR(X, P ) = 0, for all i ≥ e,
where e is a positive integer or 0}.

If X ∈ Mod-R (or R-Mod), let RejP (X) = {Ker(f ) | f ∈ HomR(X, P )}.
For every R-module P , ProdP (prodP ) will denote the class of the R-modules
isomorphic to arbitrary (finite) summands of direct products of copies of P. Add(P )
(add(P )) will denote the class of the R-modules isomorphic to arbitrary (finite)
summands of direct sums of copies of P. Cogen(P ) (cogen(P )) consists of the R-
modules that embed in arbitrary (finite) direct products of modules isomorphic to
P. An R-module N ∈ Cogen(P ) if and only if, for every 0 = x ∈ N, there is a
morphism f ∈ HomR(N, P ) such that f (x) = 0. Gen(P ) will denote the class of
the R-modules generated by P. For a module M, if there exists an exact sequence
0 → M → P X → P Y , where X, Y are sets, we say M is copresented by PR. A
subcategory is resolving if it contains all projective objects and is closed under
extensions and kernels of epimorphisms.
For a bimodule APR, we let ∆PR = HomR(−, P ) and ∆AP = HomA(−, P ). Both
of which will, when convenient, simply be denoted by ∆P or ∆. We have a pair of
contravariant functors

∆PR : Mod-R A-Mod : ∆AP .

Associated with this adjunction are the evaluation maps

δX : X → ∆2X with δX (x) : f → f (x)

r-COSTAR MODULES 169

for X in Mod-R or A-Mod, x ∈ X, and f ∈ ∆(X). This yields natural transforma-

tions

δ : 1Mod-R → ∆2 and δ : 1A-Mod → ∆2.

A module X is (P -)reflexive if δX is an isomorphism, and (P -)torsionless if δX is a

monomorphism. Note that KerδX = RejP (X), so that X is P -torsionless if and only
if X is cogenerated by P. We shall let ΓPR = ExtR1 (−, P ) and ΓAP = Ext1A(−, P ),
both of which will usually be denoted by Γ, cf. [2].

Denoting the class of P -reflexive right-R and left-A modules by Ref(PR) and
Ref(AP ), respectively, it always the case that APR-induces a duality

∆ : Ref(PR) Ref(AP ) : ∆.

Let fgd-tl(PR) denote by the class of torsionless right R-modules whose P -duals
are finitely generated over A and fg-tl(AP ) denote the class of finitely generated
torsionless left A-modules. we also denote by

sf-cp(PR) = {M ∈ Mod-R| 0 → M → P n → P A with n ∈ N }.

Colby and Fuller obtain the following theorem.

Theorem 2.1. ([1]) Let R be a ring, P ∈ Mod-R and A = End(PR). The following
are equivalent.

(a) ∆ : fgd-tl(PR) fg-tl(AP ) : ∆ is a duality. That is, PR is a costar module.
(b) ∆ : sf-cp(PR) fg-tl(AP ) : ∆ is a duality and fgd-tl(PR) = sf-cp(PR).
(c) δM is an epimorphism if ∆(M ) ∈ A-mod and δN is an epimorphism if
N ∈ A-mod.
(d) fgd-tl(PR) ⊆ sf-cp(PR) and if 0 → M →f P n → L → 0 is exact with L ∈
Cogen(PR), then ∆(f ) is an epimorphism.
(e) If 0 → M →f P n → L → 0 is exact then L ∈ Cogen(PR) if and only if ∆(f )
is an epimorphism.

A ring Λ is an artin algebra if its center K is an artinian ring and Λ is finitely
generated as a K-module. Any finitely generated module over an artin algebra is
finitely generated over its endmorphism ring, which is also an artin algebra, cf. [2].

In this paper, we let Λ denote by an artin albebra and we only consider finitely
generated module category. If P ∈ mod-Λ, for M ∈ mod-Λ(A-mod), then ∆(M ) ∈
A-mod(mod-Λ). Any exact sequence 0 → L → P n → M → 0 in mod-Λ(or A-mod),
then L, M ∈ mod-Λ(or A-mod) and ∆(L), ∆(M ) ∈ A-mod(or Λ-mod), since Λ is
an artin algebra. If Λ is an artin algebra and APΛ is a bimodule, then Cogen(PΛ)
∩ mod-Λ=cogen(PΛ) and Cogen(AP ) ∩ A-mod=cogen(AP ), cf. [2](P 115). So that

170 HONGXING LIU AND SHUNHUA ZHANG

from Theorem 2.1 (e), P is a costar module provided that any exact sequence
0 → M →f P n → L → 0 remains exact after applying the functor ∆P if and only if

L ∈ cogen(PΛ).

3. Basic properties on r-costar modules

Firstly, we give the definition of r-costar modules.

Definition 3.1. Let P ∈ mod-Λ. A Λ-module P is said to be an r-costar module
provided that any exact sequence 0 → L → M → N → 0 with L, M ∈ Ref(PΛ)
remains exact after applying the functor ∆PΛ if and only if N ∈ Ref(PΛ).

If P is an r-costar module and cogen(PΛ) = Ref(PΛ), then P is a costar module.
For an r-costar module PΛ, the subcategory Ref(PΛ) has the following properties.

Proposition 3.2. Let PΛ ∈ mod-Λ be an r-costar module. Then the following hold.
(1) The functor ∆PΛ preserves short exact sequences in Ref(PΛ).
(2) For any M ∈ Ref(PΛ), there is an infinite exact sequence 0 → M → P1 →

··· → Pn → ··· which remains exact after applying the functor ∆PΛ where Pi ∈ prodP
for each i.

(3) For any exact sequence 0 → L → M → N → 0 which is also exact after
applying the functor ∆PΛ , if two of its terms are in Ref(PΛ), then so is the third
one.

Proof. (1) Follows from the definition of r-costar modules.
(2) If X is finitely generated, then ∆(X) is finitely generated as A-module. So,

for any M ∈ Ref(PΛ), let ∆(M ) is generated by f1, · · ·, fn as A-module. Then we
have an exact sequence 0 → M →f P n → M1 → 0 where f = (f1, · · ·, fn) and
P n ∈ prodP. Since the sequence is clearly exact after applying the functor ∆PΛ and
P n ∈ Ref(PΛ), we obtain that M1 ∈ Ref(PΛ). By repeating the process to M1, and
so on, we finally obtain the desired exact sequence.

(3) If L, M ∈ Ref(PΛ), then N ∈ Ref(PΛ) by the definition of r-costar modules.
Now let N ∈ Ref(PΛ). By applying the functor ∆2 to the sequence in (3), we
have the following commutative diagram by assumptions.

0→ L → M → N →0
↓δL ↓δM ↓δN

0 → ∆2(L) → ∆2(M ) → ∆2(N )

r-COSTAR MODULES 171

since N ∈ Ref(PΛ), we have that δN is an isomorphism. It follows easily that δL is
an isomorphism if and only if δM is an isomorphism. So that, L ∈ Ref(PΛ) if and
only if M ∈ Ref(PΛ).

The following proposition gives some properties of the subcategory Ref(AP ) for
P an r-costar module.

Proposition 3.3. Let P ∈ mod-Λ be an r-cotar module with A = End(PΛ). Then
the following hold.

(1) projA ⊆ Ref(AP ) ⊆ KerExtiA≥1(−, P ). In particular, the functor ∆AP pre-
serves short exact sequences in Ref(AP ).

(2) Ref(AP ) is a resolving subcategory.

Proof. Since A ∈ Ref(AP ), we have projA ⊆ Ref(AP ). Now for any N ∈ Ref(AP ),
then ∆(N ) ∈ Ref(PΛ). By proposition 3.2(2), there is an infinite exact sequence
0 → ∆(N ) → P1 → · · · → Pn → · · · which remains exact after applying the functor
∆PΛ to the sequence. We obtain an exact sequence · · · → ∆(Pn)(=∼ An) → · · · →
∆(P1)(∼= A1) → ∆2(N ) → 0 with Ai ∈ projA. It remains exact after applying the
functor ∆AP . We obtain that Ref(AP ) ⊆ KerExtiA≥1(−, P ) by dimension shifting.

(2) From (1), we know that Ref(AP ) contains all projective A-modules. For
any exact sequence 0 → K → M → N → 0 with N ∈ Ref(AP ). As Ref(AP ) ⊆
KerExtAi≥1(−, P ) by (1), we obtain the following exact commutative diagram by
applying the functor ∆2.

0→ K → M → N →0

↓δK ↓δM ↓δN

0 → ∆2(K) → ∆2(M ) → ∆2(N ) .

Since N ∈ Ref(AP ), then δN is an isomorphism. So that, δK is an isomorphism
if and only if δM is an isomorphism. That is, K ∈ Ref(AP ) if and only if M ∈
Ref(AP ). It follows that Ref(AP ) is closed under kernels of epimorphisms and under
extensions. Thus, Ref(AP ) is a resolving subcategory.

4. Characterizations of r-costar modules

In this section, we shall give an equivalent condition for r-costar modules.

Proposition 4.1. Let P be an right Λ-module with A = End(PΛ). Assume that
Ref(AP ) ⊆ KerExtiA≥1(−, P ) and KerExtAi≥0(−, P ) = 0. Then P is an r-costar
module.

172 HONGXING LIU AND SHUNHUA ZHANG

Proof. Let 0 → L → M → N → 0 be an exact sequence with L, M ∈ Ref(PΛ).
Assume the sequence is exact after applying the functor ∆, then we have the induced

exact sequence 0 → ∆(N ) → ∆(M ) → ∆(L) → 0. Since ∆(L) ∈ Ref(AP ) ⊆
KerExtiA≥1(−, P ), after applying the functor ∆, we obtain an exact sequence 0 →
∆2(L) → ∆2(M ) → ∆2(N ) → 0. Note that L, M ∈ Ref(PΛ), so δL, δM are
isomorphisms. It follows that δN is an isomorphism and N ∈ Ref(PΛ).

Now, suppose that N ∈ Ref(PΛ). By applying the functor ∆, we obtain an
induced exact sequences 0 → ∆(N ) → ∆(M ) → X → 0 and 0 → X →f ∆(L) →

Y → 0 for some X, Y ∈ A-mod. After applying the functor ∆ to the first sequence,

we then have the following exact commutative diagram

0→ L → M → N → 0

↓h ↓δM ↓δN ↓

0 → ∆(X) → ∆2(M ) → ∆2(N ) → Ext1A(X, P ) → 0.

Since M, N ∈ Ref(PΛ), δM , δN are isomorphisms. It follows from the diagram
that ExtA1 (X, P ) = 0 and h = ∆(f )δL is an isomorphism. Then ∆(f ) is an iso-

morphism since δL is an isomorphism. Note that ∆(M ), ∆(N ) ∈ Ref(AP ) ⊆
KerExtiΛ≥1(−, P ), so we have ExtAi (X, P ) = 0 for all i ≥ 2 by dimension shifting.
Hence X ∈ KerExtAi≥1(−, P ). On the other hand, by applying the functor ∆ to the

sequence 0 → X → ∆(L) → Y → 0, we have an induced exact sequence

0 −→ ∆(Y ) −→ ∆2(L) ∆−(→f) ∆(X) −→ Ext1A(Y, P ) −→ 0

and that Y ∈ KerExtiA≥2(−, P ), since ∆(L) ∈ Ref(AP ) ⊆ KerExtAi≥1(−, P ). Hence
we obtain that Ext1A(Y, P ) = 0 = ∆(Y ). It follows that Y ∈ KerExtiA≥0(−, P ). So
that Y = 0 by assumptions and hence X ∼= ∆(L) canonically. Therefore, we deduce
that the functor ∆ preserves the exactness of the exact sequence 0 → L → M →
N → 0 in Ref(PΛ).

Finally, we conclude that P is an r-costar module.

Lemma 4.2. Let P ∈ mod-Λ and A = End(PΛ). Then Ω2(KerExtiA≥1(−, P )) ⊆
Ref(AP ) where Ω2 denotes the second syzygy module.

Proof. For any N ∈ Ω2(KerExtiA≥1(−, P )), we have an exact sequence 0 → N →
A2 → A1 → M → 0 with A1, A2 ∈ projA and M ∈ KerExtiA≥1(−, P ). We have an
induced exact sequence

0 → ∆(M ) → ∆(A1) → ∆(A2) → ∆(N ) → 0

r-COSTAR MODULES 173

by applying the functor ∆. Now after applying the functor ∆ to the last sequence,
we obtain an induced exact sequence

0 → ∆2(N ) → ∆2(A2) → ∆2(A1).

Since δA1 , δA2 are isomorphisms, we deduce that δN is also an isomorphism. Hence
N ∈ Ref(AP ).

Proposition 4.3. Let PΛ be an r-costar module. Then KerExtiA≥0(−, P ) = 0.
Proof. For any M ∈ KerExtiA≥0(−, P ), we have an exact sequence 0 → N → A2 →
A1 → M → 0 with A1, A2 ∈ projA and N ∈ Ω2(M ). Then, by applying the
functor ∆ to the sequence, we obtain an induced exact sequence 0 → ∆(A1) →
∆(A2) → ∆(N ) → 0. By Lemma 4.2, N ∈ Ref(AP ) and then ∆(N ) ∈ Ref(PΛ). So,
after applying the functor ∆, we obtain that the induced sequence

0 → ∆2(N ) → ∆2(A2) → ∆2(A1) → 0

is exact by Proposition 3.2. It follows that M = Coker(A2 → A1) =∼ Coker(∆2(A2) →
∆2(A1)) = 0.

From Propositions 3.3, 4.1 and 4.3, we have the following characterization of
r-costar modules.

Theorem 4.4. Let P ∈ mod-Λ and A = End(PΛ). Then P is an r-costar module
if and only if projA⊆ Ref(AP ) ⊆ KerExtAi≥1(−, P ) and KerExtiA≥0(−, P ) = 0.

From Theorem 4.4, we obtain the following result.

Theorem 4.5. Let P ∈ mod-Λ and A = End(PΛ). If Ref(AP ) = KerExtAi≥1(−, P ),
then P is an r-costar module.

Proof. Under assumptions we clearly have that projA ⊆ Ref(AP ). For any M ∈
KerExtAi≥0(−, P ). Then M ∈ Ref(AP ) = KerExtiA≥1(−, P ). Hence M =∼ ∆2(M ) =
0. By Theorem 4.4, we obtain that P is an r-costar module.

PL in condition (2) in the following proposition can be replaced by P n. So
that, from the following proposition, we know that r-costar modules are obtained
by replacing the subcategory cogen(PΛ) with the subcategory Ref(PΛ) in costar
modules theory under artin algebra situation.

Proposition 4.6. Let P ∈ mod-Λ. The following are equivalent.
(1) P is an r-costar module.
(2) For any exact sequence 0 → L → PL → M → 0 with PL ∈ prodP and L ∈

174 HONGXING LIU AND SHUNHUA ZHANG

Ref(PΛ), then M ∈ Ref(PΛ) if and only if the functor ∆ preserves the exactness of
the sequence.

Proof. (1) ⇒ (2) is followed from the definition of r-costar module.
(2) ⇒ (1) As the proof in Proposition 3.2(2), for any L ∈ Ref(PΛ), we have

an infinite exact sequence 0 → L → P1 → · · · → Pn → · · · which remains exact
after applying the functor ∆, where Pi ∈ prodP for each i. Consequently, we have
projA⊆ Ref(AP ) ⊆ KerExtiA≥1(−, P ) as in the proof of Proposition 3.3(1).

At last, the same proof as in Proposition 4.3 yields that KerExtiA≥0(−, P ) = 0.
Thus P is an r-costar module by Theorem 4.4.

5. r-Costar modules with special properties

In this section, we shall study r-costar modules P such that the subcategory
Ref(PΛ) has some special properties.

Proposition 5.1. Let P ∈ mod-Λ with A = End(PΛ). If AP is an injective cogen-
erator in A-mod, then PΛ is an r-costar module.

Proof. Let 0 → L → M → N → 0 be an exact sequence with L, M ∈ Ref(PΛ).
Assume first that the sequence is exact after applying the functor ∆, then we have
an induced exact sequence 0 → ∆(N ) → ∆(M ) → ∆(L) → 0. Since AP is injective,
after applying the functor ∆, we obtain that the sequence 0 → ∆2(L) → ∆2(M ) →
∆2(N ) → 0 is exact. Note that L, M ∈ Ref(PΛ), so, δL and δM are isomorphisms.
It follows that δN is an isomorphism and N ∈ Ref(PΛ).

Now, suppose that N ∈ Ref(PΛ). After applying the functor ∆, we obtain an
induced exact sequence 0 → ∆(N ) → ∆(M ) → ∆(L) → D → 0 for some D. Since
AP is injective, we have an exact sequence 0 → ∆(D) → ∆2(L) → ∆2(M ) →
∆2(N ) → 0 by applying the functor ∆. Because δL, δM , δN are isomorphisms,
then ∆(D) = 0. As AP is a cogenerator, we then obtain that D = 0. Hence
0 → ∆(N ) → ∆(M ) → ∆(L) → 0 is exact. Therefore, P is an r-costar module.

For an r-costar module P, we know that Ref(AP ) is a resolving subcategory from
Proposition 3.3(2). Now we consider when Ref(PΛ) is a resolving subcategory. The
following theorem gives some characterizations of this case.

Theorem 5.2. Let P ∈ mod-Λ. The following are equivalent.
(1) P is an r-costar module such that Ref(PΛ) is a resolving subcategory.
(2) Ref(PΛ) = KerExtΛi≥1(−, P ).

r-COSTAR MODULES 175

(3) Ref(PΛ) ⊆ KerExtΛi≥1(−, P ) ⊆ cogen(PΛ).
(4) projΛ ⊆ Ref(PΛ) ⊆ KerExtiΛ≥1(−, P ) and projA⊆ Ref(AP ) ⊆ KerExtiA≥1(−, P ).

Proof. (1) ⇒ (2) Let M ∈ Ref(PΛ) and 0 → M1 → Λ1 → M → 0 be an exact se-

quence with Λ1 ∈ projΛ. Since Ref(PΛ) is a resolving subcategory, projΛ ⊆ Ref(PΛ)
and Ref(PΛ) is closed under kernels of epimorphisms. Therefore M1 ∈ Ref(PΛ).

It follows from Proposition 3.2 that the sequence remains exact after applying the
functor ∆. Then we have that Ext1Λ(M, P ) = 0 for any M ∈ Ref(PΛ). Hence,
Ref(PΛ) ⊆ KerExtiΛ≥1(−, P ), since Ref(PΛ) is a resolving subcategory.

On the other hand, we may consider the exact sequence 0 → X → Λ2 → Λ1 →
M → 0 with Λ1, Λ2 ∈ projΛ for any M ∈ KerExtΛi≥1(−, P ). It clearly remains
exact after applying the functor ∆. Hence we have an induced exact sequence 0 →

∆(M ) → ∆(Λ1) → ∆(Λ2) → ∆(X) → 0. Since Ref(PΛ) is a resolving subcategory,

then projΛ ⊆ Ref(PΛ). By applying the functor ∆ to the sequence, we obtain that
X =∼ ∆2(X). That is, X ∈ Ref(PΛ). Since P is an r-costar modules, then we

have that Im(Λ2 → Λ1) ∈ Ref(PΛ) and similarly M ∈ Ref(PΛ). Hence we have
KerExtΛi≥1(−, P ) ⊆ Ref(PΛ).

(2) ⇒ (3) is clear.
(3) ⇒ (4) Since prodP ⊆ Ref(PΛ), we have that prodP ⊆ KerExtΛi≥1(−, P ).
For any M ∈ KerExtiΛ≥1(−, P ), let ∆(M ) is generated by f1, · · ·, fn as A-module.
Then we have an exact sequence 0 → M →f P n → M1 → 0 where f = (f1, · ·

·, fn). Note that the sequence clearly stays exact after applying the functor ∆ and
since M, P n ∈ KerExtΛi≥1(−, P ), we obtain that M1 ∈ KerExtΛi≥1(−, P ) too. By
repeating the process to M1, and so on, we finally obtain an infinite exact sequence

0 → M → P1 → · · · → Pn → · · · with Pi ∈ prodP for each i, such that the

sequence remains exact after applying the functor ∆. It follows that M ∈ Ref(PΛ).
Hence we deduce that Ref(PΛ) = KerExtΛi≥1(−, P ), therefore projΛ ⊆ Ref(PΛ).
Moreover, by an argument similar to the proof of Proposition 3.3(1), we obtain
that projA ⊆ Ref(AP ) ⊆ KerExtAi≥1(−, P ).

(4) ⇒ (1) Since projΛ ⊆ Ref(PΛ) ⊆ KerExtiΛ≥1(−, P ), we easily check that
Ref(PΛ) is a resolving subcategory as the proof in Proposition 3.3(2). Now it

remains to show that P is an r-costar module. By Theorem 4.4, we need only
to prove that KerExtAi≥0(−, P ) = 0. Let M ∈ KerExtiA≥0(−, P ) and take an exact
sequence 0 → N → A2 → A1 → M → 0 with A2, A1 ∈ projA. Then we have
an induced exact sequence 0 → ∆(A1) → ∆(A2) → ∆(N ) → 0 by applying the

functor ∆. Note that N ∈ Ref(AP ) by Lemma 4.2, so that ∆(N ) ∈ Ref(PΛ) ⊆
KerExtiΛ≥1(−, P ). It follows that there is an induced exact sequence 0 → ∆2(N ) →

176 HONGXING LIU AND SHUNHUA ZHANG

∆2(A2) → ∆2(A1) → 0 by applying the functor ∆. Hence we obtain that M =
Coker(A2 → A1) ∼= Coker(∆2(A2) → ∆2(A1)) = 0.

Proposition 5.3. Let P be an r-costar module. Then Ref(PΛ) is closed under
extensions if and only if Ext1Λ(M, P ) = 0 for any M ∈ Ref(PΛ).

Proof. ⇒ . Every exact sequence of the form 0 → P →f N → M → 0 where
M ∈ Ref(PΛ). By hypothesis, N ∈ Ref(PΛ). Then ∆ preserves the exactness of the
sequence by the definition of r-costar modules. So that, we have g : N → P such
that gf = 1P . Hence the sequence splits and ExtΛ1 (M, P ) = 0.

⇐ . Let any exact sequence 0 → L → M → N → 0 with L, N ∈ Ref(PΛ). Then
∆ preserves the exactness of the sequence by assumption. We have the following
commutative diagram

0→ L → M → N → 0
↓δL ↓δM ↓δN ↓

0 → ∆2(L) → ∆2(M ) → ∆2(N ) → X → 0.

Note that δL, δN are isomorphisms, since L, N ∈ Ref(PΛ). Therefore, we have that
δM is an isomorphism. That is, M ∈ Ref(PΛ).

References

[1] R. R. Colby and K. R. Fuller, Costar modules, J. Algebra, 242 (2001), 146-159.
[2] R. R. Colby and K. R. Fuller, Equivalence and Duality for Module Categories:

with tilting and cotilting for rings, Cambridge tracts in mathematics : 161,
Cambridge university press, 2004.
[3] R. Colpi, Some remarks on equivalences between categories of modules, Comm.
Algebra, 18(6) (1990), 1935-1951.
[4] R. Colpi and C. Menini,On the structure of ∗-modules, J. Algebra, 158 (1993),
400-419.
[5] C. Menini and A. Orsatti, Reprsentable equivalences between categories of mod-
ules and applications, Rend. Sem. Mat. Univ. Padova, 82 (1989), 203-231.
[6] J. Trlifaj, Every ∗-module is finitely generated, J. Algebra, 169 (1994), 392-398.
[7] J. Wei, ∗s-Modules, J. Algebra, 291 (2005), 312-324.

Hongxing Liu Shunhua Zhang
School of Mathematics Science School of Mathematics
Shandong Normal University Shandong University
250014 Jinan, P. R. China 250100 Jinan, P. R. China
e-mail: [email protected] e-mail: [email protected]